Universal properties of Fermi gases in one dimension

In this Rapid Communication, we investigate the universal properties of a spin-polarized two-component Fermi gas in one dimension (1D) using the Bethe ansatz. We discuss the quantum phases and phase transitions by obtaining exact results for the equation of state, the contact, the magnetic susceptibility, and the contact susceptibility, giving a precise understanding of the 1D analog of the Bose-Einstein condensation and Bardeen-Cooper-Schrieffer crossover in three dimensions (3D) and the associated universal magnetic properties. In particular, we obtain the exact form of the magnetic susceptibility $\chi \sim 1/\sqrt{T}exp(-\mathrm{\Delta }/T)$ at low temperatures, where $\mathrm{\Delta }$ is the energy gap and $T$ is the temperature. Moreover, we establish exact upper and lower bounds for the relation between polarization $P$ and the contact $C$ for both repulsive and attractive Fermi gases. Our findings emphasize the role of pair fluctuations in strongly interacting 1D fermion systems that can shed light on higher dimensions.

Figure 1

The dimensionless pressure ratio of $p/p_0$ vs $\mu /T$ for an 1D Fermi gas. Here, $p_0$ is the pressure for the free fermions. (a) shows the numerical TBA result for the EOS of the polarized Fermi gas with repulsive (lower branch) and attractive (upper branch) interactions for different magnetic fields [50]. (b) The rescaled effective chemical potential of bound pairs ${\mu }_2/{\epsilon }_F$ as a function of $ln\left|\gamma \right|$ for $H=0$. (c) At high temperatures or weak coupling, the curves for this ratio show agreement between the analytical result Eq. (1) (circles) and the numerical TBA result (solid lines) for $H=0$. (d) The ratio $p/p_0$ vs $\mu /T$ for the strongly attractive Fermi gas with different interacting strengths: Solid lines stand for the numerical TBA result while the circles denote the analytical result from Eq. (3).

Figure 2

Dimensionless susceptibility vs temperature for the attractive gas with a chemical potential $\mu =-0.249$ and a coupling strength $c=-1$. For the external field $h<h_{c1}=0.996$, the susceptibility of Eq. (4) (thick black-dashed lines) is in excellent agreement with the numerical result from the TBA equations [50]. The red dotted line shows the onset of susceptibility at $h=h_{c1}$. For $h>h_{c1}$ the gap vanishes with an antiferromagnetic behavior.

Figure 3

The dimensionless contact $\tilde{C}$ and contact susceptibility ${\tilde{C}}_h$ as a function of dimensionless temperature $t$ for the attractive Fermi gas with $\tilde{\mu }=-0.498$. (a) and (c) $\tilde{C}$ and ${\tilde{C}}_h$ for different magnetic fields near the upper critical field $h_{c2}=1.0675$. (b) and (d) $\tilde{C}$ and ${\tilde{C}}_h$ for different magnetic fields near the lower critical field $h_{c1}=0.9962$. (e) and (f) show the contour plots of the contact susceptibility as a function of the temperature at the upper and the lower critical points, respectively. The black lines show the minima of the contact susceptibility.

Figure 4

Contact ratio $C/C_{\max }$ vs polarization $P$. Here, $C_{\max }$ is the corresponding maximum value of the contact at $P=0$. Left panel: For an attractive interaction with $\gamma =-10$. Right panel: For a repulsion with $\gamma =100$. The ratios in the two interacting regimes are bound between the two limiting cases, viz., $1-P^2$ and $1-P$.